arXiv subjects

Aritra Mitra

Publications and source records attributed to Aritra Mitra.

At least 37 records · Page 2Linked to original sources

arXiv2024

Robust Q-Learning under Corrupted Rewards

Recently, there has been a surge of interest in analyzing the non-asymptotic behavior of model-free reinforcement learning algorithms. However, the performance of such algorithms in non-ideal environments, such as in the presence of corrupted rewards, is poorly understood. Motivated by this gap, we investigate the robustness of the celebrated Q-learning algorithm to a strong-contamination attack model, where an adversary can arbitrarily perturb a small fraction of the observed rewards. We start by proving that such an attack can cause the vanilla Q-learning algorithm to incur arbitrarily large errors. We then develop a novel robust synchronous Q-learning algorithm that uses historical reward data to construct robust empirical Bellman operators at each time step. Finally, we prove a finite-time convergence rate for our algorithm that matches known state-of-the-art bounds (in the absence of attacks) up to a small inevitable $O(\varepsilon)$ error term that scales with the adversarial corruption fraction $\varepsilon$. Notably, our results continue to hold even when the true reward distributions have infinite support, provided they admit bounded second moments.

cs.LG↗

arXiv2024

DASA: Delay-Adaptive Multi-Agent Stochastic Approximation

We consider a setting in which $N$ agents aim to speedup a common Stochastic Approximation (SA) problem by acting in parallel and communicating with a central server. We assume that the up-link transmissions to the server are subject to asynchronous and potentially unbounded time-varying delays. To mitigate the effect of delays and stragglers while reaping the benefits of distributed computation, we propose \texttt{DASA}, a Delay-Adaptive algorithm for multi-agent Stochastic Approximation. We provide a finite-time analysis of \texttt{DASA} assuming that the agents' stochastic observation processes are independent Markov chains. Significantly advancing existing results, \texttt{DASA} is the first algorithm whose convergence rate depends only on the mixing time $τ_{mix}$ and on the average delay $τ_{avg}$ while jointly achieving an $N$-fold convergence speedup under Markovian sampling. Our work is relevant for various SA applications, including multi-agent and distributed temporal difference (TD) learning, Q-learning and stochastic optimization with correlated data.

cs.AI↗

arXiv2024

Finite-Time Analysis of Asynchronous Multi-Agent TD Learning

Recent research endeavours have theoretically shown the beneficial effect of cooperation in multi-agent reinforcement learning (MARL). In a setting involving $N$ agents, this beneficial effect usually comes in the form of an $N$-fold linear convergence speedup, i.e., a reduction - proportional to $N$ - in the number of iterations required to reach a certain convergence precision. In this paper, we show for the first time that this speedup property also holds for a MARL framework subject to asynchronous delays in the local agents' updates. In particular, we consider a policy evaluation problem in which multiple agents cooperate to evaluate a common policy by communicating with a central aggregator. In this setting, we study the finite-time convergence of \texttt{AsyncMATD}, an asynchronous multi-agent temporal difference (TD) learning algorithm in which agents' local TD update directions are subject to asynchronous bounded delays. Our main contribution is providing a finite-time analysis of \texttt{AsyncMATD}, for which we establish a linear convergence speedup while highlighting the effect of time-varying asynchronous delays on the resulting convergence rate.

cs.MA↗

arXiv2024

Federated Temporal Difference Learning with Linear Function Approximation under Environmental Heterogeneity

We initiate the study of federated reinforcement learning under environmental heterogeneity by considering a policy evaluation problem. Our setup involves $N$ agents interacting with environments that share the same state and action space but differ in their reward functions and state transition kernels. Assuming agents can communicate via a central server, we ask: Does exchanging information expedite the process of evaluating a common policy? To answer this question, we provide the first comprehensive finite-time analysis of a federated temporal difference (TD) learning algorithm with linear function approximation, while accounting for Markovian sampling, heterogeneity in the agents' environments, and multiple local updates to save communication. Our analysis crucially relies on several novel ingredients: (i) deriving perturbation bounds on TD fixed points as a function of the heterogeneity in the agents' underlying Markov decision processes (MDPs); (ii) introducing a virtual MDP to closely approximate the dynamics of the federated TD algorithm; and (iii) using the virtual MDP to make explicit connections to federated optimization. Putting these pieces together, we rigorously prove that in a low-heterogeneity regime, exchanging model estimates leads to linear convergence speedups in the number of agents.

cs.LG↗

arXiv2024

A Simple Finite-Time Analysis of TD Learning with Linear Function Approximation

We study the finite-time convergence of TD learning with linear function approximation under Markovian sampling. Existing proofs for this setting either assume a projection step in the algorithm to simplify the analysis, or require a fairly intricate argument to ensure stability of the iterates. We ask: \textit{Is it possible to retain the simplicity of a projection-based analysis without actually performing a projection step in the algorithm?} Our main contribution is to show this is possible via a novel two-step argument. In the first step, we use induction to prove that under a standard choice of a constant step-size $α$, the iterates generated by TD learning remain uniformly bounded in expectation. In the second step, we establish a recursion that mimics the steady-state dynamics of TD learning up to a bounded perturbation on the order of $O(α^2)$ that captures the effect of Markovian sampling. Combining these pieces leads to an overall approach that considerably simplifies existing proofs. We conjecture that our inductive proof technique will find applications in the analyses of more complex stochastic approximation algorithms, and conclude by providing some examples of such applications.

cs.LG↗

arXiv2024

Temporal Difference Learning with Compressed Updates: Error-Feedback meets Reinforcement Learning

In large-scale distributed machine learning, recent works have studied the effects of compressing gradients in stochastic optimization to alleviate the communication bottleneck. These works have collectively revealed that stochastic gradient descent (SGD) is robust to structured perturbations such as quantization, sparsification, and delays. Perhaps surprisingly, despite the surge of interest in multi-agent reinforcement learning, almost nothing is known about the analogous question: Are common reinforcement learning (RL) algorithms also robust to similar perturbations? We investigate this question by studying a variant of the classical temporal difference (TD) learning algorithm with a perturbed update direction, where a general compression operator is used to model the perturbation. Our work makes three important technical contributions. First, we prove that compressed TD algorithms, coupled with an error-feedback mechanism used widely in optimization, exhibit the same non-asymptotic theoretical guarantees as their SGD counterparts. Second, we show that our analysis framework extends seamlessly to nonlinear stochastic approximation schemes that subsume Q-learning. Third, we prove that for multi-agent TD learning, one can achieve linear convergence speedups with respect to the number of agents while communicating just $\tilde{O}(1)$ bits per iteration. Notably, these are the first finite-time results in RL that account for general compression operators and error-feedback in tandem with linear function approximation and Markovian sampling. Our proofs hinge on the construction of novel Lyapunov functions that capture the dynamics of a memory variable introduced by error-feedback.

cs.LG↗

arXiv2024

Finite-Time Analysis of On-Policy Heterogeneous Federated Reinforcement Learning

Federated reinforcement learning (FRL) has emerged as a promising paradigm for reducing the sample complexity of reinforcement learning tasks by exploiting information from different agents. However, when each agent interacts with a potentially different environment, little to nothing is known theoretically about the non-asymptotic performance of FRL algorithms. The lack of such results can be attributed to various technical challenges and their intricate interplay: Markovian sampling, linear function approximation, multiple local updates to save communication, heterogeneity in the reward functions and transition kernels of the agents' MDPs, and continuous state-action spaces. Moreover, in the on-policy setting, the behavior policies vary with time, further complicating the analysis. In response, we introduce FedSARSA, a novel federated on-policy reinforcement learning scheme, equipped with linear function approximation, to address these challenges and provide a comprehensive finite-time error analysis. Notably, we establish that FedSARSA converges to a policy that is near-optimal for all agents, with the extent of near-optimality proportional to the level of heterogeneity. Furthermore, we prove that FedSARSA leverages agent collaboration to enable linear speedups as the number of agents increases, which holds for both fixed and adaptive step-size configurations.

cs.LG↗

arXiv2024

Stochastic Approximation with Delayed Updates: Finite-Time Rates under Markovian Sampling

Motivated by applications in large-scale and multi-agent reinforcement learning, we study the non-asymptotic performance of stochastic approximation (SA) schemes with delayed updates under Markovian sampling. While the effect of delays has been extensively studied for optimization, the manner in which they interact with the underlying Markov process to shape the finite-time performance of SA remains poorly understood. In this context, our first main contribution is to show that under time-varying bounded delays, the delayed SA update rule guarantees exponentially fast convergence of the \emph{last iterate} to a ball around the SA operator's fixed point. Notably, our bound is \emph{tight} in its dependence on both the maximum delay $τ_{max}$, and the mixing time $τ_{mix}$. To achieve this tight bound, we develop a novel inductive proof technique that, unlike various existing delayed-optimization analyses, relies on establishing uniform boundedness of the iterates. As such, our proof may be of independent interest. Next, to mitigate the impact of the maximum delay on the convergence rate, we provide the first finite-time analysis of a delay-adaptive SA scheme under Markovian sampling. In particular, we show that the exponent of convergence of this scheme gets scaled down by $τ_{avg}$, as opposed to $τ_{max}$ for the vanilla delayed SA rule; here, $τ_{avg}$ denotes the average delay across all iterations. Moreover, the adaptive scheme requires no prior knowledge of the delay sequence for step-size tuning. Our theoretical findings shed light on the finite-time effects of delays for a broad class of algorithms, including TD learning, Q-learning, and stochastic gradient descent under Markovian sampling.

cs.LG↗

arXiv2023

Min-Max Optimization under Delays

Delays and asynchrony are inevitable in large-scale machine-learning problems where communication plays a key role. As such, several works have extensively analyzed stochastic optimization with delayed gradients. However, as far as we are aware, no analogous theory is available for min-max optimization, a topic that has gained recent popularity due to applications in adversarial robustness, game theory, and reinforcement learning. Motivated by this gap, we examine the performance of standard min-max optimization algorithms with delayed gradient updates. First, we show (empirically) that even small delays can cause prominent algorithms like Extra-gradient (\texttt{EG}) to diverge on simple instances for which \texttt{EG} guarantees convergence in the absence of delays. Our empirical study thus suggests the need for a careful analysis of delayed versions of min-max optimization algorithms. Accordingly, under suitable technical assumptions, we prove that Gradient Descent-Ascent (\texttt{GDA}) and \texttt{EG} with delayed updates continue to guarantee convergence to saddle points for convex-concave and strongly convex-strongly concave settings. Our complexity bounds reveal, in a transparent manner, the slow-down in convergence caused by delays.

cs.LG↗

arXiv2023

Federated TD Learning over Finite-Rate Erasure Channels: Linear Speedup under Markovian Sampling

Federated learning (FL) has recently gained much attention due to its effectiveness in speeding up supervised learning tasks under communication and privacy constraints. However, whether similar speedups can be established for reinforcement learning remains much less understood theoretically. Towards this direction, we study a federated policy evaluation problem where agents communicate via a central aggregator to expedite the evaluation of a common policy. To capture typical communication constraints in FL, we consider finite capacity up-link channels that can drop packets based on a Bernoulli erasure model. Given this setting, we propose and analyze QFedTD - a quantized federated temporal difference learning algorithm with linear function approximation. Our main technical contribution is to provide a finite-sample analysis of QFedTD that (i) highlights the effect of quantization and erasures on the convergence rate; and (ii) establishes a linear speedup w.r.t. the number of agents under Markovian sampling. Notably, while different quantization mechanisms and packet drop models have been extensively studied in the federated learning, distributed optimization, and networked control systems literature, our work is the first to provide a non-asymptotic analysis of their effects in multi-agent and federated reinforcement learning.

cs.LG↗

arXiv2022

Collaborative Linear Bandits with Adversarial Agents: Near-Optimal Regret Bounds

We consider a linear stochastic bandit problem involving $M$ agents that can collaborate via a central server to minimize regret. A fraction $α$ of these agents are adversarial and can act arbitrarily, leading to the following tension: while collaboration can potentially reduce regret, it can also disrupt the process of learning due to adversaries. In this work, we provide a fundamental understanding of this tension by designing new algorithms that balance the exploration-exploitation trade-off via carefully constructed robust confidence intervals. We also complement our algorithms with tight analyses. First, we develop a robust collaborative phased elimination algorithm that achieves $\tilde{O}\left(α+ 1/\sqrt{M}\right) \sqrt{dT}$ regret for each good agent; here, $d$ is the model-dimension and $T$ is the horizon. For small $α$, our result thus reveals a clear benefit of collaboration despite adversaries. Using an information-theoretic argument, we then prove a matching lower bound, thereby providing the first set of tight, near-optimal regret bounds for collaborative linear bandits with adversaries. Furthermore, by leveraging recent advances in high-dimensional robust statistics, we significantly extend our algorithmic ideas and results to (i) the generalized linear bandit model that allows for non-linear observation maps; and (ii) the contextual bandit setting that allows for time-varying feature vectors.

cs.LG↗

arXiv2022

A Survey of Graph-Theoretic Approaches for Analyzing the Resilience of Networked Control Systems

As the scale of networked control systems increases and interactions between different subsystems become more sophisticated, questions of the resilience of such networks increase in importance. The need to redefine classical system and control-theoretic notions using the language of graphs has recently started to gain attention as a fertile and important area of research. This paper presents an overview of graph-theoretic methods for analyzing the resilience of networked control systems. We discuss various distributed algorithms operating on networked systems and investigate their resilience against adversarial actions by looking at the structural properties of their underlying networks. We present graph-theoretic methods to quantify the attack impact, and reinterpret some system-theoretic notions of robustness from a graph-theoretic standpoint to mitigate the impact of the attacks. Moreover, we discuss miscellaneous problems in the security of networked control systems which use graph-theory as a tool in their analyses. We conclude by introducing some avenues for further research in this field.

eess.SY↗

arXiv2022

Distributed Statistical Min-Max Learning in the Presence of Byzantine Agents

Recent years have witnessed a growing interest in the topic of min-max optimization, owing to its relevance in the context of generative adversarial networks (GANs), robust control and optimization, and reinforcement learning. Motivated by this line of work, we consider a multi-agent min-max learning problem, and focus on the emerging challenge of contending with worst-case Byzantine adversarial agents in such a setup. By drawing on recent results from robust statistics, we design a robust distributed variant of the extra-gradient algorithm - a popular algorithmic approach for min-max optimization. Our main contribution is to provide a crisp analysis of the proposed robust extra-gradient algorithm for smooth convex-concave and smooth strongly convex-strongly concave functions. Specifically, we establish statistical rates of convergence to approximate saddle points. Our rates are near-optimal, and reveal both the effect of adversarial corruption and the benefit of collaboration among the non-faulty agents. Notably, this is the first paper to provide formal theoretical guarantees for large-scale distributed min-max learning in the presence of adversarial agents.

cs.LG↗

arXiv2022

Linear Stochastic Bandits over a Bit-Constrained Channel

One of the primary challenges in large-scale distributed learning stems from stringent communication constraints. While several recent works address this challenge for static optimization problems, sequential decision-making under uncertainty has remained much less explored in this regard. Motivated by this gap, we introduce a new linear stochastic bandit formulation over a bit-constrained channel. Specifically, in our setup, an agent interacting with an environment transmits encoded estimates of an unknown model parameter to a server over a communication channel of finite capacity. The goal of the server is to take actions based on these estimates to minimize cumulative regret. To this end, we develop a novel and general algorithmic framework that hinges on two main components: (i) an adaptive encoding mechanism that exploits statistical concentration bounds, and (ii) a decision-making principle based on confidence sets that account for encoding errors. As our main result, we prove that when the unknown model is $d$-dimensional, a channel capacity of $O(d)$ bits suffices to achieve order-optimal regret. To demonstrate the generality of our approach, we then show that the same result continues to hold for non-linear observation models satisfying standard regularity conditions. Finally, we establish that for the simpler unstructured multi-armed bandit problem, $1$ bit channel-capacity is sufficient for achieving optimal regret bounds. Overall, our work takes a significant first step towards paving the way for statistical decision-making over finite-capacity channels.

cs.LG↗

arXiv2021

Exploiting Heterogeneity in Robust Federated Best-Arm Identification

We study a federated variant of the best-arm identification problem in stochastic multi-armed bandits: a set of clients, each of whom can sample only a subset of the arms, collaborate via a server to identify the best arm (i.e., the arm with the highest mean reward) with prescribed confidence. For this problem, we propose Fed-SEL, a simple communication-efficient algorithm that builds on successive elimination techniques and involves local sampling steps at the clients. To study the performance of Fed-SEL, we introduce a notion of arm-heterogeneity that captures the level of dissimilarity between distributions of arms corresponding to different clients. Interestingly, our analysis reveals the benefits of arm-heterogeneity in reducing both the sample- and communication-complexity of Fed-SEL. As a special case of our analysis, we show that for certain heterogeneous problem instances, Fed-SEL outputs the best-arm after just one round of communication. Our findings have the following key implication: unlike federated supervised learning where recent work has shown that statistical heterogeneity can lead to poor performance, one can provably reap the benefits of both local computation and heterogeneity for federated best-arm identification. As our final contribution, we develop variants of Fed-SEL, both for federated and peer-to-peer settings, that are robust to the presence of Byzantine clients, and hence suitable for deployment in harsh, adversarial environments.

cs.LG↗

arXiv2021

Linear Convergence in Federated Learning: Tackling Client Heterogeneity and Sparse Gradients

We consider a standard federated learning (FL) architecture where a group of clients periodically coordinate with a central server to train a statistical model. We develop a general algorithmic framework called FedLin to tackle some of the key challenges intrinsic to FL, namely objective heterogeneity, systems heterogeneity, and infrequent and imprecise communication. Our framework is motivated by the observation that under these challenges, various existing FL algorithms suffer from a fundamental speed-accuracy conflict: they either guarantee linear convergence but to an incorrect point, or convergence to the global minimum but at a sub-linear rate, i.e., fast convergence comes at the expense of accuracy. In contrast, when the clients' local loss functions are smooth and strongly convex, we show that FedLin guarantees linear convergence to the global minimum, despite arbitrary objective and systems heterogeneity. We then establish matching upper and lower bounds on the convergence rate of FedLin that highlight the effects of intermittent communication. Finally, we show that FedLin preserves linear convergence rates under aggressive gradient sparsification, and quantify the effect of the compression level on the convergence rate. Our work is the first to provide tight linear convergence rate guarantees, and constitutes the first comprehensive analysis of gradient sparsification in FL.

cs.LG↗

arXiv2021

On the Computational Complexity of the Secure State-Reconstruction Problem

In this paper, we discuss the computational complexity of reconstructing the state of a linear system from sensor measurements that have been corrupted by an adversary. The first result establishes that the problem is, in general, NP-hard. We then introduce the notion of eigenvalue observability and show that the state can be reconstructed in polynomial time when each eigenvalue is observable by at least $2s+1$ sensors and at most $s$ sensors are corrupted by an adversary. However, there is a gap between eigenvalue observability and the possibility of reconstructing the state despite attacks - this gap has been characterized in the literature by the notion of sparse observability. To better understand this, we show that when the $\mathbf{A}$ matrix of the linear system has unitary geometric multiplicity, the gap disappears, i.e., eigenvalue observability coincides with sparse observability, and there exists a polynomial time algorithm to reconstruct the state provided the state can be reconstructed.

eess.SY↗

arXiv2021

Distributed Inference with Sparse and Quantized Communication

We consider the problem of distributed inference where agents in a network observe a stream of private signals generated by an unknown state, and aim to uniquely identify this state from a finite set of hypotheses. We focus on scenarios where communication between agents is costly, and takes place over channels with finite bandwidth. To reduce the frequency of communication, we develop a novel event-triggered distributed learning rule that is based on the principle of diffusing low beliefs on each false hypothesis. Building on this principle, we design a trigger condition under which an agent broadcasts only those components of its belief vector that have adequate innovation, to only those neighbors that require such information. We prove that our rule guarantees convergence to the true state exponentially fast almost surely despite sparse communication, and that it has the potential to significantly reduce information flow from uninformative agents to informative agents. Next, to deal with finite-precision communication channels, we propose a distributed learning rule that leverages the idea of adaptive quantization. We show that by sequentially refining the range of the quantizers, every agent can learn the truth exponentially fast almost surely, while using just $1$ bit to encode its belief on each hypothesis. For both our proposed algorithms, we rigorously characterize the trade-offs between communication-efficiency and the learning rate.

eess.SY↗